The problem is to solve the rational equation $\frac{16}{3x+2} - \frac{4x+1}{3x^2+8x+4} = \frac{4}{x+2}$ for $x$. If there is no solution, we must state that.

1. Problem Description

The problem is to solve the rational equation $\frac{16}{3x+2} - \frac{4x+1}{3x^2+8x+4} = \frac{4}{x+2}$ for $x$. If there is no solution, we must state that.

2. Solution Steps

First, we factor the quadratic expression $3x^2+8x+4$.

$3x^2+8x+4 = (3x+2)(x+2)$

Now, we can rewrite the equation as:

$\frac{16}{3x+2} - \frac{4x+1}{(3x+2)(x+2)} = \frac{4}{x+2}$

We multiply both sides by $(3x+2)(x+2)$ to eliminate the fractions:

$16(x+2) - (4x+1) = 4(3x+2)$

$16x + 32 - 4x - 1 = 12x + 8$

$12x + 31 = 12x + 8$

Subtract $12x$ from both sides:

$31 = 8$

Since $31 = 8$ is a contradiction, there is no solution to the equation.

We should also check for extraneous solutions by ensuring that $x \ne -\frac{2}{3}$ and $x \ne -2$. Since we arrived at a contradiction, there are no solutions.

3. Final Answer

B. The equation has no solution.